The Argentinian's 53rd-minute strike proved the difference as Pep Guardiola's side made it five successive wins in all competitions, matching its best run of victories in 2019-2020.
Defenders Nicolas Otamendi and Benjamin Mendy struck the woodwork either side of half-time before Championship side Wednesday's resistance was eventually broken.
Goalkeeper Joe Wildsmith got a hand to Aguero's strike but could not prevent the ball going over the line as City's all-time leading scorer registered his 254th goal for the club.
Gabriel Jesus and Aguero were both in the City starting line-up and the former headed a 20th-minute chance wide from a John Stones cross.
Aguero spotted his fellow forward's run seven minutes later but the Brazilian failed to get a good connection on a stretching volley when free six yards out.
Otamendi nodded against the crossbar from a free-kick delivered by Riyad Mahrez, who also picked out Stones from a set-piece only for the defender to miss the target when his downward header hit the ground and bounced over the top.
There was a greater impetus from City after the break, though, as Mendy's first-time effort cannoned off the frame of the goal in a sign Guardiola's half-time words had worked.
Wednesday did not heed the warning, with Aguero left in enough space in the box to collect Mendy's pass, turn and fire a shot which looped off Wildsmith's left arm and went in.
Replays showed it was a close call over whether Aguero was onside or had strayed beyond last man Dominic Iorfa, though the absence of VAR at Wednesday's ground meant there was no closer inspection.
Wildsmith did redeem himself somewhat by thwarting Stones, Rodri and substitute Raheem Sterling with his legs, while another more conventional save with his hands denied Mahrez.
However, Wednesday failed to register a shot on target across the piece and mustered their only effort overall in the sixth minute as Aguero's strike proved enough to send City into the last eight.